1
POPULATION INERTIA AND ITS SENSITIVITY TO CHANGES IN
VITAL RATES OR INITIAL CONDITIONS
David N. Koons1, 2, Randall R. Holmes3, James B. Grand4
1
Alabama Cooperative Fish and Wildlife Research Unit, School of Forestry and Wildlife
Sciences, Auburn University, AL 36849-5418
2
Max Planck Institute for Demographic Research, Konrad-Zuse Str. 1, D-18057 Rostock,
Germany
3
Department of Mathematics and Statistics, College of Science and Mathematics, Auburn
University, AL 36849-5310
4
USGS Alabama Cooperative Fish and Wildlife Research Unit, School of Forestry and
Wildlife Sciences, Auburn University, AL 36849-5418
Current contact information for corresponding author:
David N. Koons
Max Planck Institute
Konrad-Zuse Str. 1
D-18057 Rostock
Germany
e-mail: Koons@demogr.mpg.de
phone: 49-381-2081-226
fax: 49-381-2081-526
2
Abstract. Many studies have examined Keyfitz’s population momentum, a
special case of inertia in long-term population size resulting from demographic transition
to the stationary population growth rate. Yet, population inertia can be produced by any
demographic perturbation (i.e., not just perturbations that produce stationary growth).
Insight into applied population dynamics, population ecology, and life history evolution
has been gained using perturbation analysis of the population growth rate. However, a
similar, generalized framework for perturbation analysis of population inertia has not
been developed. We derive general formulas for the sensitivity of population inertia to
change in any vital rate or initial population structure. These formulas are readily
computable, and we provide examples of their potential use in life history and applied
studies of populations.
Key words: age; inertia; life history; population momentum; population structure;
sensitivity; stable equivalent ratio.
1. Introduction
Population size is central to the fields of demography and population biology.
Demographers often study population size because it can affect economies, policy, social
dynamics, and even natural resource supplies (Bos et al., 1994; Fischer and Heilig, 1997;
United Nations, 2003). Biologists pay special attention to population size when trying to
understand ecological processes, keep small populations from going extinct, controlling
pest populations, and in management of populations that provide hunting, fishing, and
viewing opportunities as well as world-market food resources (Caughley, 1977).
Additionally, the change in population size over time (i.e., population growth rate)
3
describes the average fitness and performance of the population (Fisher, 1930; Sibly et
al., 2002). Thus, population size is an important parameter for many reasons.
In population modeling it is common to assume a stable population structure (i.e.,
the distribution of abundance across age, stage, size, sex, or other phenotypic categories)
because it greatly simplifies study of how underlying vital rates, such as fecundity and
survival, affect population size and growth. Yet, if a population has an unstable
population structure, such as an ‘over abundance’ of mature adults, it will experience
transient dynamics (i.e., unstable short-term dynamics; Fox and Gurevitch, 2000; Koons
et al., 2005; Yearsley, 2004). Unstable population structure and transient dynamics can
in turn create an inertial effect on long-term population size causing it to be larger or
smaller than that of a population of the same initial size but with stable population
structure and growing according to the same vital rates (Tuljapurkar and Lee, 1997). The
most commonly studied case of population inertia is Keyfitz’s heralded concept of
population momentum (1971a), which pertains to the special case when a population’s
vital rates undergo a change to the stationary level (i.e., the level of lifetime individual
replacement) (see Fig. 1).
Because population size can affect a variety of political, social, and economic
issues, population momentum has been studied extensively in human demography and its
effect has been shown to occur in many populations (e.g., Keyfitz, 1971a; Frauenthal,
1975; Mitra, 1976; Wachter, 1988; Schoen and Kim, 1991; Fischer and Heilig, 1997;
Kim and Schoen, 1997; Schoen and Kim, 1998; Bongaarts and Bulatao, 1999; Li and
Tuljapurkar, 1999, 2000; Schoen and Jonsson, 2003). Through simulation, biologists
have recently shown that population momentum could affect conservation efforts of
4
declining wild populations and control of pest populations (Koons et al., 2006), that it
varies with life history in a predictable pattern (Koons et al., In Press), and that it is an
important aspect of optimal harvest theory (Hauser et al., In Review).
Like population growth rate, population inertia is as relevant to population
biology as it is to classical human demography, and tools that relate change in underlying
demographic parameters to change in population inertia could benefit a variety of
demographic disciplines. The sensitivity of population growth rate to changes in
underlying vital rates has a long history of use in demography and evolutionary theory
(Lewontin 1965, Hamilton 1966, Demetrius 1969, Emlen 1970, Goodman 1971, Keyfitz
1971b, Mertz 1971), and Caswell’s (1978) discrete-time sensitivity formula has made
calculation of this metric relatively simple for empiricists to use (e.g., van Groenendael et
al., 1988; Horvitz et al., 1997; Benton and Grant, 1999; papers within Heppell et al.,
2000). Analogous analytical formulae for population inertia would be just as useful, but
have not been developed. Here, we present such formulae for the sensitivity of
population inertia to changes in any vital rate or initial conditions. We then show how
these formulae can be used to examine dynamic underpinnings of population inertia
across life histories, and in applied population analysis.
2. Derivation
2.1. Population model and notation
We use bold-type capital letters to denote matrices and bold-type lower case
letters to denote vectors. We use x to denote the conjugate of x, xT to denote the
transpose of the vector x and x* to denote the complex conjugate transpose.
5
Most studies of population inertia are conducted with continuous time models.
However, the underlying mechanisms of population inertia are more easily seen in
discrete form (Schoen and Jonsson, 2003). Thus, our derivation is based on a linear,
discrete, time-invariant population model
n t+1  An t .
(1)
Here, n t is an n-dimensional vector with ith entry n i (t ) equal to the number of
individuals in the ith stage at time t. A is an n × n matrix with (i, j)-entry aij equal to the
transition rate from the jth stage to the ith stage. Alternatively, the population vector at
any time t can be expressed as:
n
n t   ci λ it w i ,
(2)
i1
where the  i ’s are the eigenvalues of A (which we assume to be distinct), the wi ’s are
corresponding right eigenvectors, and the ci’s depend on initial conditions and satisfy
n 0   i1 c i w i (Caswell, 2001). The ith eigenvalue  i and corresponding right wi and
n
left vi eigenvectors of A satisfy
Aw i  λ i w i
(3)
v i A  λ i v i .
(4)
Unless otherwise stated, we assume that A is primitive so that, according to the PerronFrobenius theorem (Gantmacher, 1959; Seneta, 1981; Horn and Johnson, 1985), there is a
unique eigenvalue of A having modulus strictly larger than the moduli of the other
eigenvalues. The indexing is chosen in such a way that  1 is this “dominant”
eigenvalue. It is real and positive, and both w1 and v1 have real positive entries. For
6
large t, the i = 1 term dominates the expression for n t given in equation 2, and so
eventually, the population grows approximately geometrically at the rate  1 (assuming
c1  0 ). The dominant right w1 and left v1 eigenvectors describe the asymptotic
population structure and reproductive values, respectively (Goodman, 1968). Unless
otherwise noted, we assume that the eigenvectors have been scaled so that v*i w i  1 , and
v*i w j  0 for i  j . The 1-norm of the vector x is given by x   i 1 xi , where xi is the
n
ith component of x.
2.2. Measurement of population inertia
In a deterministic environment, population inertia is defined as the long-term size
of a population growing at any rate as determined by the actual population structure,
relative to the size of an otherwise equivalent population that grows according to its
stable population structure (Fig. 1b). In demography, this ratio is known as the Stable
Equivalent Ratio (SER; sensu Tuljapurkar and Lee, 1997). In order to relate various
notions of population inertia and momentum that have appeared in the literature, we
consider the quantity
M r0 lim
t 
n t r0
,
rt n 0
(5)
which we call the “inertia relative to r0 ”. Here r0 is a nonzero vector and rt  At r0 .
We regard r0 as a point of reference and therefore call it the “reference vector”. Since
n t r0
rt n 0
At

n0
n0
r
At 0
r0
,
(6)
7
one can see that M r0 is the asymptotic ratio of the size of the population with initial
structure n 0 normalized to a unit vector to that with initial structure r0 normalized to a
unit vector. Note that M r0 depends only on the line determined by r0 , that is,
M cr0  M r0 for any nonzero complex number c.
Here, we assume that the eigenvalues of A are indexed in such a way that
1   2 
  n . Moreover, any vector x can be written in the form x   i 1 di w i
n
for some uniquely determined complex numbers di . If x is nonzero, then di is nonzero
for some i and we call the least such i the “height” of x. Note that the height of x is also
the least index i for which v*i x is nonzero.
In Appendix 1, we establish that
M r0 
v *h n 0 r0
v *h r0 n 0
,
(7)
provided  i   h for i > h, where h is the lesser of the height of n 0 and the height of
r0 (or the common value if they are the same). The fraction is interpreted as being
infinity if the denominator is zero.
Now we consider various special cases of M r0 that are of notable biological
interest. First, assume that r0  cw 1 for some positive real number c. Since, equation 5
gives, in this case, the Stable Equivalent Ratio (SER) described above, we set
SER = M r0  M cw1 . From remarks above, we see that this quantity does not depend on
c. Because r0 has height one, equation 7 yields
8
SER 
v 1*n 0 r0
(8)
v 1*r0 n 0
(noting that the absolute value signs can be dropped since the vectors on the right are all
real and nonnegative). If we further assume that r0 is of the same size as n 0 (i.e.,
r0  n0 ), then equation 8 reduces to
SER 
v 1*n 0
,
v 1*r0
(9)
which is consistent with equation 7 in Tuljapurkar and Lee (1997).
Next, by replacing r0 in equation 8 with cw 1 and canceling c’s we get
SER 
v 1*n 0 w 1
v 1*w 1
n0



e T v 1*n 0 w 1
T
e n0
,
(10)
where e is the vector with each entry equal to one. This formula is the same as Caswell’s
discrete-time formula for population momentum (Caswell, 2001:104). Caswell considers
momentum resulting from instantaneous demographic changes to the stationary level,
that is, where  1  1 . In this case, rt  cw1  r0 for all t so equation 5 gives
SER K  lim
t 
nt
n0
 assuming  1  1 ,
(11)
which is equivalent to Keyfitz’s (1971a) general definition of population momentum
(hence the subscript K on SER). This, in turn, is a special case of population inertia
(Schoen and Kim, 1991; Tuljapurkar and Lee, 1997; Schoen and Jonsson, 2003; Keyfitz
and Caswell, 2005). Caswell derives equation 10 from equation 11 making use of the
assumption  1  1 . Our derivation of equation 10 from equation 5 does not require this
assumption.
9
Lastly, if the initial vector n 0 has height > 1, so that v1*n0  0 (we momentarily
relax the assumption that A is primitive to allow for this possibility), then the SER is zero
and all that can be concluded is that the size of the population will become increasingly
less significant as compared to that of a stable population growing at rate  1 . However,
the population might still be viable with, say, its structure vector approaching an
eigenvector for a subdominant eigenvalue. In this case, the SER can be regarded as being
too coarse a measure to distinguish between this population and one that, for instance, is
moving toward extinction, which would also clearly have zero SER. We get a more
refined view of inertial effects by choosing various reference vectors r0 in equation 5.
For instance, if n 0 has height h > 1 and we let r0  w h and assume that  i   h for i >
h, then equation 7 gives
M r0 
v *hn 0 w h
n0
.
(12)
This inertia M r0 is nonzero (since v *hn 0  0 ) and it gives the asymptotic size of the
given population relative to the size of a stable population growing at the subdominant
rate  h . In the remainder of this paper we focus on the SER and leave further
investigation of this more general notion of inertia to future work.
The SER in equations 9 and 10 can be used to measure inertia in population size
caused by an initially unstable population structure, or inertia resulting from
instantaneous change in any vital rate to a new level (i.e., a ‘demographic transition’).
For example, n 0 can represent the initial population structure or it can represent the
population structure produced by a historical set of vital rates. Of course, following a
10
‘demographic transition’, v 1 , w 1 , and r0 would refer to the post-transition set of vital
rates. Henceforth, we use the terms ‘population inertia’ and SER interchangeably.
2.3. General formulas for the sensitivity of population inertia to changes in vital rates
A may represent the single set of focal vital rates, or, following a demographic
transition, A may represent the set of post-transition vital rates. In both cases, one might
be interested in the question: “how would population inertia change if the vital rates were
just a little bit different.” A measure for the sensitivity of population inertia to change in
the underlying vital rates (aij) of the transition matrix A is needed to answer this question.
Thus, to develop general formulas for such a sensitivity measure, we begin with
equation 10 because this form of the deterministic SER should be most familiar to readers
(e.g., presented as an equation for the more familiar M in Caswell, 2001; Keyfitz and
Caswell, 2005). To begin, we apply the product rule to differentiate equation 10 with
respect to a single vital rate aij:
 eT 
 SER
  
=
aij
aij 

 v1*n0  w1  



eT n 0
 T 

 v*n w   0 
e

1 0
1
 
eT n0  aij 

w1  
1  T *
e  v1n0
=

v1*n 0
T

aij  aij
e n0  
=

1

=



(13)

 
 w1  
 
 
 
w1  v1*   
eT  v1*n0

n0  w1   .
 
aij  aij
eT n0  
 
1


11
It becomes apparent that perturbation of a vital rate causes change in the right w 1 and
left v 1 eigenvectors. Conveniently, Caswell (1980, 2001) has developed formulas for
sensitivities of these eigenvectors to change in a vital rate aij:
n
vi(m)
w1
 w(1)
wm ,
j 
aij



1
m
m 1
(14)
n
w(jm)
v1
(1)
 vi 
vm.
aij



1
m
m 1
(15)
where w(jm ) is the jth entry of w m and vi( m) is the ith entry of v m . By incorporating
Caswell’s formulas (our eqns. 14 and 15) into equation 13, the sensitivity of the
deterministic SER to change in a vital rate can be seen as
 SER
1

aij
eT n 0
*
 

 
( m)
( m)

n
n

 
w
 T *
v
 
j
(1)
i
w m     vi(1) 
v m  n0  w1   . (16)
e  v1n0  w j 

 

  m
 
m 1 1
m 1 1   m
 

 


 
 


Thus, change in a vital rate causes change in the stable population vector w 1 and
reproductive value vector v 1 , which then leads to change in the SER. In appendix 2, we
prove that equations 14 and 15 can indeed be incorporated into equation 13 to yield
equation 16.
Sometimes matrix-level entries are computed from multiple lower-level vital
rates. For example, projection matrix fertilities are the product of fecundity and some
component of survival. The sensitivity of population inertia to a lower-level parameter x
can be found using equation 16 together with the chain rule
 SER
 SER akl

.
x
x
k ,l akl
(17)
12
Population inertia sensitivities can also be calculated numerically
 SER
aij
SER post  SER pre
ij
.
(18)
Here, SER pre and SER post are the measurements of population inertia before and after
the instantaneous change  ij in the vital rate aij . Although we have chosen to focus on
the sensitivity of population inertia to absolute change in vital rates, the elasticity of
population inertia to proportional change in vital rates can easily be calculated from the
analytical sensitivity or numerically,
 logSER  SER aij

 log aij
aij SER
 logSER
 log aij
SER post  SER pre 1
SER pre
pij
(19)
(20)
where pij is the proportional change in the vital rate aij . Population inertia is not a
linear function of the aij , thus the elasticities do not sum to unity nor do they quantify the
contribution of the aij to SER like they do for the geometric population growth rate (de
Kroon et al., 1986). Still, elasticities do provide useful measures of the effect of relative
change in a vital rate on population inertia.
2.4. General formulas for the sensitivity of population inertia to changes in population
structure
Population inertia also depends on the initial population vector n 0 , and one may
be interested in how changes in initial conditions cause change in population inertia.
Thus, we develop general formulas for this sensitivity as well. To begin, we apply the
13
quotient rule to differentiate equation 10 with respect to a single entry ni (0) of the initial
population vector:
 SER
  v1*n0 
T
= e w1


ni (0)
ni (0)  eT n0 




 T

*
*
e
n
v
n

v
n
eTn 0 
0
1
0
1
0

ni (0)
ni (0)

= eT w1 
2


T
e n0



 

(21)
 T

(1)
e n0 v i  v1*n 0 

= eT w1 
.
2
T


e n0




We note that the ‘initial’ point in time can be defined as the point in time from which the
population will be studied forward. We assume for the rest of this section that
eT w1  w1  1, eTn0  n0  1 , and v1 is real. Then
 SER
= vi (1) - v1T n0 .
ni (0)
(22)
In particular, if one examines the special case where n0  w1 , then
 SER
= vi (1) -1.
ni (0) w
1
(23)
Thus, in equation 22, knowledge of only the reproductive value and initial population
structure are needed to measure the sensitivity of population inertia to change in a single
entry of the initial population structure. If it is safe to assume that the population
structure is initially stable (eqn. 23), then only reproductive value is needed, which is
easily computed from A.
14
Now we consider perturbations that could affect multiple (st)age classes. To do
this, let u  u1,
, u n  be an arbitrary unit vector (i.e., u  1 ) to be regarded as a
T
perturbation vector applied to n 0 . Furthermore, denote by DuSER the directional
derivative of SER in the direction u. Then
DuSER = SER T u,
where SER =  SER n1(0),
(24)
,  SER nn (0)  is the gradient of the SER. The set
T
of all vectors x for which SER T x  0 forms a hyperplane in n-space. The set of those x
for which SER T x  0 (respectively, SER T x  0 ) are on the positive side
(respectively, negative side) of the hyperplane. Thus, the directional derivative is
positive, negative, or zero when u is on the positive side of the hyperplane, the negative
side, or in the hyperplane itself, respectively.
Considering the special case where n 0  w 1 (with each being a unit vector), we
get
DuSER w = SER Tu
1
(25)
=  v1  e  u.
T
In this case the SER equals 1. Therefore, the new value of population inertia after
perturbation is > 1, < 1, or = 1 when the directional derivative is > 0, < 0, or = 0,
respectively. If one perturbs only the ith entry of w 1 (i.e., u  0,
,0,1,0,
,0 ), then
T
the sign of the ith entry of v1  e dictates the direction that the SER moves away from 1.
In this case
15
 SER
 DuSER
= vi (1) -1.
ni (0) w
1
w1
(26)
Thus, equation 23 is just a special case of the directional derivative (eqn. 25). In
addition, if u  w 1 , then the distribution of individuals among stage classes does not
change following the perturbation (u), so the SER is expected to remain at 1. This is
indeed the case, since Dw1 SER
  v1  e  w1  v1T w1  eT w1  v1T w1  1  0 .
T
w1
3. Applications and examples
3.1. Age-related dynamics of inertia in an open population
Nearly all studies of population inertia in demography have focused on
momentum following change in fertility to the stationary level. Yet, changes in agestructured vital rates other than fertility are expected to produce population inertia and
warrant more study (Li and Tuljapurkar, 1999). Furthermore, spatial structuring of
populations can significantly affect population momentum, and more generally,
population inertia (Rogers and Willekens, 1978; Rogers, 1995). In addition to fertility
and survival, dynamics of ‘open’ populations are affected by immigration and
emigration. Here, we provide a short example illustrating how our formulae can be used
to examine and compare how population inertia is affected by changes in each of the ageor stage-structured vital rates of an open population.
It is relatively straightforward to incorporate net immigration into projection
models (e.g., Rogers, 1995), but Cooch et al. (2001) present a concise model for
incorporating birth, survival, immigration, and emigration vital rates into a single
projection matrix:
16
0

S   I
 x x x
0
A



0
0
0
0
mx S00
0
0
0
S x x  I x
mx S00 
0 
0 


0 
where A is constructed in the traditional pre-breeding census format (Caswell, 2001), m x
is the average number of births per female in age- or stage-class x, S0 is the probability
of surviving from birth to census in age or stage 1, S x is the survival probability from
age- or stage-class x to x + 1,  0 and  x are the corresponding probabilities of remaining
in the population, conditional on being alive (i.e., 1 – emigration probability, which is
called fidelity), and I x is the probability that an individual present in the population at
time t in age- or stage-class x + 1 was not present in the population at time t - (census
time) in age or stage-class x (i.e., the probability of immigrating during the time between
censuses from outside the local population). In the example presented here, we
parameterized this model (A) with data based on a female segment of the U.S. population
< 50 years old, counted at 5-year intervals, and growing by 1% per year (Table 1). Data
were based on the 1980, U.S. Southwest life tables presented in Rogers (1995), but
adjusted to fit our example described above. We assumed that the population initially
had a stable population structure and then applied equations 16, 17, and 18 to calculate
the elasticity of population inertia (i.e., the SER) to changes in each of the
aforementioned vital rates.
Of great importance, the sign of the survival, fidelity (  x ), and immigration
elasticities changed from negative to positive with increasing age while the fecundity
elasticity was always negative, indicating that increased fecundity always created inertia
17
leading to smaller population size relative to an otherwise equivalent population in its
stable population structure (Fig. 2). (We note that elasticity values indicate results for an
increase in a vital rate, and decreases would produce exactly the opposite result. We also
note that elasticities for survival probabilities and fidelity were identical because of their
perfect multiplicative relation in the model). The effect of changing a vital rate on the
population structure and reproductive value (see eqn. 13) varied with age, leading to the
patterns in SER elasticities across age (Fig. 2). Furthermore, changes in fecundity did not
have the largest impact on the SER, survival and local population fidelity did. Thus,
while demographers seem to be quite concerned about the effects of the fertility transition
on population momentum (see numerous citations in Introduction), perhaps they should
also be concerned about the effects of increasing longevity (sensu Guillot, 2005) and
changes in migration rates on the SER, a more general measure of population inertia.
Moreover, population inertia affects abundance in the various age classes, which
is known to have profound impacts on the economy (Lee, 2000) and use of natural
resources (Liu et al., 1999, 2003). In addition, population abundance can be a measure of
evolutionary fitness for populations regulated by density (Grant and Benton, 2000).
Although our model does not include density-dependence, inertia can occur in these
populations. Thus, our resultant patterns in the elasticity of population inertia to changes
in vital rates across age (see Fig. 2) could be very important in each of the
aforementioned situations. From our example, it is quite apparent that change in a given
vital rate for age-class 30 would have a very different effect on population inertia, and
thus abundance, than would the same change in age-class 5 (Fig. 2).
18
Age-related patterns in population inertia sensitivities and elasticities may be a
general result of significant importance. Although this is just one simple example,
further study of population inertia sensitivities and elasticities across age and stage are as
warranted as the numerous studies of population growth rate sensitivities, elasticities, and
selection pressures. Furthermore, population inertia sensitivities and elasticities could
prove to be very useful for studying the dynamics of multi-regional populations (Rogers,
1995), and meta-populations (Hanski and Gilpin, 1997) such as source-sink systems (e.g.,
Koons, 2005).
3.2. Life history and the sensitivity of population inertia
The study of patterns in population dynamics across populations and species is
important because it aids in the understanding of life history evolution (Harvey and
Pagel, 1991; Stearns, 1992), and in the development of demographic policy as well as
conservation and management of populations (Bos et al., 1994; Fagan et al., 2001).
Adding to the repertoire of methods used in comparative demography, our formulas
allow one to compare the functional relationship between vital rates and population
inertia across populations and across species. Furthermore, the analytical formulae
provide a consistent means for comparison that alleviate some pitfalls of numerical
simulation.
To provide a brief example of such an application, we used the bird data with
stationary growth (  1  1 ) provided in Koons et al. (In Press). Population models were
constructed as in Koons et al. (In Press), and we assumed that each population initially
had a stable population structure and stationary growth, hence the SER = 1. This served
as a nice starting point because any perturbation to these equilibrium conditions produces
19
a SER ≠ 1, and it is this ‘change’ in population inertia that we were interested in
comparing across species with different life history attributes. Specifically, we applied
equation 16 to the population model (A) for each species and calculated the sensitivity of
the SER to changes in fertility, sub-adult survival, and adult survival.
Across the avian life histories, we found that the sensitivity of the SER to changes
in fertility increased with the life history generation time (p < 0.01, R2 = 0.94, Fig. 3a) as
did sensitivity of the SER to changes in sub-adult survival (p < 0.01, R2 = 0.86, Fig. 3b).
Interestingly, sensitivity of the SER to change in sub-adult survival was negative for
species with a generation time < 10 but positive for species that mature later in life and
have a longer generation time. Thus, whether changes in sub-adult survival produced
inertia leading to an enlarged or reduced population size relative to an otherwise
equivalent population in the stable population structure depended on the duration of the
sub-adult stage. On the other hand, sensitivity of the SER to change in adult survival
decreased with life history generation time and was always negative (Fig. 3c). Although
statistical support for the latter relationship was weak (p = 0.12, R2 = 0.27), this brief
analysis indicates that the functional relationship between vital rates and population
inertia does tend to vary with life history, which is consistent with Koons et al’s more indepth numerical analysis of population momentum across vertebrate life histories (In
Press). Our equations can be used for analysis following a population’s transition to
stationary growth, which would be of interest when studying population momentum, but
can also be used to study non-equilibrium and non-stationary conditions as well. We
suggest use of these new tools to examine population inertia dynamics in a wide variety
of populations and species.
20
3.3. Effects of initial population structure
In many cases, managers might want to consider how different management
strategies focused on population structure could change population inertia in their favor
when managing population abundance (MacArthur, 1960; Merrill et al., 2003; Hauser et
al., In Review). Plant and animal release and relocation programs provide managers a
variety of ways to directly ‘add’ individuals to specific age or stage classes of a
population (e.g., Starling, 1991; Wolf et al., 1996; Ostermann et al., 2001), while harvest,
live-trapping, and other removal techniques allow managers to directly decrease
abundance in specific age or stage classes (Larkin, 1977; Holt and Talbot, 1978). All of
these management practices could change population inertia. Thus, we provide an
example that illustrates how population inertia is affected by perturbations that add or
remove individuals from specific age classes of a population. We use the following
matrix A, which describes the mean fertility and survival rates of the lesser snow goose
(Chen caerulescens) population at La Perouse Bay, Manitoba from 1973 to 1990 (Cooch
et al., 2001).
0.12 0.26 0.38 0.41
 0
0.83
0
0
0
0 

A 0
0.83
0
0
0 


0
0.83
0
0 
 0
 0
0
0
0.83 0.83
In A, fertilities are represented on the top row and survival probabilities are on the subdiagonal and bottom-right corner of the matrix. The 1st age class represents young and all
others represent adult age classes.
21
To begin, we calculated the left v1  0.34 0.44 0.52 0.56 0.57 and right
T
w1  0.46 0.36 0.28 0.21 0.73 eigenvectors of A. To simplify our scenario, we
T
assumed that population structure was initially stable (i.e., n 0  w 1 ), indicating SER
initially = 1. We also normalized w1 into a unit vector (i.e., w1, nor.  1 ), and
normalized v1 , such that v1,nor.  v1 w1 , which allowed us to use equations 25 and 26
to easily address our scenario. In addition, the normalized right and left eigenvectors are
still eigenvectors of A, and the condition v*i w i  1 is maintained. Rounded to the second
decimal, w1, nor.  0.23 0.17 0.14 0.10 0.36 and
T
v1, nor.  0.70 0.90 1.06 1.15 1.17 .
T
Next, we used the directional derivative to measure the sensitivity of population
inertia to a variety of perturbations to initial population structure (Table 2). It is readily
seen that if only the ith entry of initial population structure is perturbed, then the direction
SER moves away from 1 is dictated by the sign of the ith entry of SER or vi (1) -1 . For
example, a unit increase in the 1st age class decreases population inertia (SER < 1), a unit
increase in an older adult age class increases population inertia (SER > 1), and unit
decreases produce opposite results (Table 2). Furthermore, perturbations to multiple age
classes produce intermediate, but perhaps more realistic results. Perturbations equal to
the stable age distribution ( w1, nor. ) do not change population inertia at all (SER still = 1;
Table 2).
Compared to the stable age distribution of A, removing young (e.g., through
clutch removal), releasing adults, or both, would allow a goose manager to quickly
22
increase population inertia because each action shifts age structure towards reproducing
adults. On the other hand, releasing young, removing adults (via harvest or live
trapping), or both, would allow the manager to quickly decrease population inertia
because these practices shift age structure towards young. However, it is important to
remember that these results pertain only to the effects of directly changing population
structure, not the effects of perturbing vital rates, which are presented above.
In general, the influence of population structure on eventual population size is of
great importance in conservation biology and management (Keyfitz and Caswell, 2005).
When reintroducing species into previously habited areas, it would be most effective to
introduce individuals of an age and reproductive value that would maximize the SER (see
Koons, 2005). Similarly, invasion of non-native plants and animals can lead to
substantial economic and environmental damage. Management aimed towards
minimizing the SER would be of interest in the management of invasive species (Keyfitz
and Caswell, 2005; Koons et al., 2006) and our formulae could help refine management
agendas in these situations.
4. Closing Remarks
Population inertia is a measure related to population size rather than growth rate,
and unlike the growth rate, population size is very responsive to initial population
structure (Tuljapurkar and Lee, 1997). Direct or indirect changes in population structure
create transient dynamics having an effect on both short- and long-term size, which is an
issue that is best addressed with formal perturbation measures like sensitivities.
Sensitivities have been developed for population size (Fox and Gurevitch, 2000;
Yearsley, 2004). However, we have drawn upon theory developed by Caswell (1980) to
23
derive a suite of sensitivity formulae for the SER that explicitly show that population
inertia depends on both reproductive value and population structure (Tuljapurkar and
Lee, 1997; eqn. 13). These formulae differ from those for population size itself because
the SER is a ratio measuring the inertial effect of population structure and transient
dynamics on long-term population size relative to an otherwise equivalent population in
its asymptotically stable population structure.
Most often, demographers and population biologists refer to asymptotic measures
of population dynamics (e.g., r,  1 , T) making the SER especially useful because it
provides a direct comparison of the dynamics resulting from changes in the vital rates and
population structure to the stable population dynamics (Fig. 1b). Thus, the SER
sensitivities can readily be used to examine the consequences of assuming a stable
population structure, which is common practice. Furthermore, we have shown that SER
sensitivities could be used to address a large array of new questions in human
demography, population biology, evolution, conservation, and natural resource
management as well as many other arenas.
Formulas relating change in underlying vital rates to change in the asymptotic
population growth rate (e.g., Caswell 1978), transient dynamics (Fox and Gurevitch,
2000; Caswell, 2001; Yearsley, 2004; Koons et al. 2005), and inertia in long-term
population size (here) are all needed to better understand population dynamics from
evolutionary and applied points of view. We encourage further exploration of the
behavior of population inertia in nature, as well as theoretical work on the maximization,
minimization, and control of population inertia (e.g., Koons, 2005; Hauser et al., In
Review).
24
Acknowledgments
The authors thank R.F. Rockwell, F.S. Dobson, B. Zinner, G.R. Hepp, M.C.
Wooten, and several anonymous individuals for helpful comments and review of an
earlier version of this paper. DNK and JBG appreciate the support of D.V. Derksen and
Lyman K. Thorsteinson of USGS, Biological Resources Division. This research
addresses a priority identified by the Minerals Management Service, Alaska Outer
Continental Shelf Region. The USGS, Biological Resources Division, Alaska Biological
Science Center funded this research. DNK was supported by a Max Planck Society postdoctoral fellowship in the final stages of this research.
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31
Appendix 1.
Here, we establish equation 7, namely,
M r0 
v *h n 0 r0
v *h r0 n 0
,
under the assumption  i   h for i > h, where h is the lesser of the height of n 0 and
the height of r0 . We can write n0   i h ci w i and r0  i h di wi with either ch  0 or
n
n
dh  0 . We have, using equation 2,
 ih ci it w i
n
 ih d i it w i
n
M r0
n0
r0
 lim
t 
=
nt
rt
chw h
d hw h
 lim
t 

ch
dh

v *h n 0
v *h r0
 i h ci  i
n
 i h d i  i
n
 lim
t 
h  w i
t
h  w i
t
,
and the equality follows. For the fourth equality, we have used the fact that for i > h,
 i   h , implying limt   i h   0 . Note also that if dh  0, then ch  0 , so the
t
limit is ∞. In this case, v*hr0  0 , so the final fraction is interpreted as being ∞ in
accordance with our stated convention.
32
Appendix 2.
We discuss in detail some technical issues regarding our derivation of the
population inertia sensitivity formula presented in equation 16 of the text. To begin, fix a
pair (i, j) and view the matrix A as a function of t  aij alone (this dependence is
indicated by writing A = A(t)). Let I be an open interval of real numbers such that A(t)
is irreducible for all t  I and fix t0  I . We claim that the dominant right w 1 and left
v 1 eigenvectors of A can be chosen in such a way that
(A1) w 1 and v 1 are both differentiable functions of t on the interval I ,
(A2)
w1(t0 ), w1(t0 )  1,
(A3)
w1' , v1  0 and w1, v1'  0,
(A4)
w1, v1  1.
Here, the prime symbol ( ' ) denotes derivative with respect to t and the equations in A3
and A4 are understood to be identities, that is, valid for all t  I . Also, for vectors v and
w we are writing, for clarity, v, w instead of v*w . (Reasons for choosing the
eigenvectors to satisfy A1-A4 are given in the remarks below.)
We first establish a general result.
Lemma: Let x and y be n-dimensional real vector functions on an open real interval I .
Assume that x is differentiable, x' and y are continuous, and x(t ), y(t )  0 for all t  I .
Let c0 and t0 be real numbers with t0  I and c0 > 0. There exists a differentiable
positive real-valued function c on I such that (cx)', y  0 and c(t0 )  c0 .
33
Proof: We have (cx) ', y  c' x, y  c x' , y , so the desired function c is a solution to
the initial value problem c'  pc  0 , c(t0 )  c0 , where p  x' , y
x, y . This has a
t
unique solution, namely c(t )  c0 exp   p(u ) du  , which is evidently positive.
 t0

ˆ 1 (t ) and vˆ 1 (t ) denote the unique positive right and left
For t  I , let w
ˆ 1(t ), w
ˆ 1(t )  1 and
eigenvectors of A(t) corresponding to  1 (t ) such that w
vˆ 1(t ), vˆ 1(t )  1. This is guaranteed by our assumption that A(t) is irreducible for all
t  I . Then ŵ1 and v̂1 are continuously differentiable vector functions on I .
ˆ 1, y  vˆ 1, c0  1. Then w1  cw
ˆ 1 , with c
Fix t0  I . Apply the lemma with x  w
as in the lemma, is a differentiable vector function satisfying w1' , vˆ 1  0 and
ˆ 1 (t0 ). Apply the lemma again, but this time with x  vˆ 1, y  w1,
w1 (t0 )  w
ˆ 1(t0 ), vˆ 1(t0 )
c0  w
1
. Then v1  cvˆ 1 , with c as in the lemma, is a differentiable vector
ˆ 1(t0 ), vˆ 1(t0 )
function satisfying v1' , w1  0 and v1(t0 )  w
1
vˆ 1(t0 ). Note that
'
w1' , v1  w1' , cvˆ 1  c w1' , vˆ 1  0. Now w1, v1  w1' , v1  w1, v1'  0 , so w1, v1
ˆ 1 (t0 ), w
ˆ 1 (t0 ), vˆ 1 (t0 )
does not depend on t. Since w1 (t0 ), v1 (t0 )  w
1
vˆ 1 (t0 )  1 , it
follows that w1, v1  1 on I . This establishes the claim that w 1 and v 1 can be chosen
to satisfy A1-A4.
Remarks: In view of A1 and A4, we can use w 1 and v 1 in equation 9 to view the SER as
a differentiable function of t on the interval I . Writing the differential dw1  w1' dt in the
34
form dw1   m1 am w m as in Caswell (2001:249) and taking the inner product with v 1
s
of both sides, we see that a1  0 since w1' , v1  0 (by A3). Therefore, the derivation
Caswell gives of the right eigenvector sensitivity formula (eqn. 14 in our paper) is valid
for this choice of w 1 and v 1 (with the remaining eigenvectors chosen arbitrarily). Since
w1, v1'  0 (by A3), the left eigenvector sensitivity formula (eqn. 15) is valid as well.
Finally, we point out that if a computer program automatically scales the eigenvectors so
that w1, w1  1 and w1, v1  1 (as does Matlab), then, in view of A2 and A4, it gives
the desired results when applied to our SER sensitivity formula in equation 16 at any
fixed parameter value aij  t0  I .
35
Table 1. Values of survival probability (S), local population fidelity (  ), immigration
probability (I) and fecundity (m, female births per female between and including age x
and x + 4) for the female segment of the U.S. Southwest population < 50 years old,
counted at 5-year intervals, and growing by 1 % per year.
Vital Ratesa
Age x
S

I
m
0
0.938
0.956
-
0
5
0.971
0.973
0.087
0.0008
10
0.977
0.979
0.076
0.0978
15
0.968
0.974
0.102
0.2880
20
0.956
0.963
0.144
0.3698
25
0.955
0.962
0.144
0.2760
30
0.962
0.970
0.104
0.1313
35
0.969
0.978
0.065
0.0389
40
0.970
0.985
0.042
0.0067
45
0.966
0.988
0.030
0.0004
a – Data were based on life tables presented in Rogers (1995), but adjusted to fit our example described in
the text.
36
Table 2. The sensitivity of population inertia to unit changes u in the initial population
structure of the lesser snow goose population at La Perouse Bay, Manitoba, indicated by
the directional derivative DuSER w , and the effect of the perturbation on the SER. We
1
used the normalized left eigenvector v1, nor.  0.70 0.90 1.06 1.15 1.17 , and the
T
following gradient of the SER: SER   0.30 0.10 0.06 0.15 0.17  , each
T
rounded to the second decimal.
ua
DuSER w
1
[1 0 0 0 0]T
-0.30
<1
[0 0 1 0 0]T
0.06
>1
[0 0 0 0 1]T
0.17
>1
[-1 0 0 0 0]T
0.30
>1
[0 0 -1 0 0]T
-0.06
<1
[0 0 0 0 -1]T
-0.17
<1
[0.2 0.2 0.2 0.2 0.2]T
-0.003
<1
[-0.2 -0.2 -0.2 -0.2 -0.2]T
0.003
>1
[0.23 0.17 0.14 0.10 0.36]T b
0
a – Rounded to the second decimal
b – The stable age distribution
Value of SER after perturbation
1
37
Figure 1. Graphical examples of a) Keyfitz’s “population momentum” showing
increased population size caused by an unstable population structure (solid line) relative
to the size of an otherwise equivalent population growing according to its stable
population structure and a stationary growth rate (dashed line), and b) “population
inertia”, which is essentially the same phenomenon, but not restricted in definition to
populations that eventually attain the stationary population growth rate (e.g., a growing
population is shown here). Measures of population inertia are given in section 2.2.
Figure 2. The elasticity of the Stable Equivalent Ratio (SER) to changes in
survival probability, local population fidelity, immigration probability, and fecundity
across age categories for the U.S. Southwest population example (section 3.1).
Figure 3. The relationship between life history generation time and the sensitivity
of the Stable Equivalent Ratio (SER) to changes in a) fertility, b) sub-adult survival, and
c) adult survival for initially stationary populations of selected avian species. The scale
of the y-axes differs among graphs a, b, and c. (Life history data comes from Koons et al.
In Press).
38
a)
280
260
Population Size
240
220
200
180
160
"Population Momentum"
140
120
100
80
0
5
10
15
20
25
30
Time
b)
450
400
Population Size
350
300
"Population Inertia"
250
200
150
100
50
0
5
10
15
Time
20
25
30
39
0.3
Elasticity of SER to
change in vital rate
0.2
0.1
0.0
-0.1
-0.2
Survival
Fidelity
Immigration
Fecundity
-0.3
-0.4
-0.5
0
5
10
15
20
25
Age Category
30
35
40
45
40
a)
1.0
Sensitivity of SER
to change in fertility
0.8
0.6
0.4
0.2
0.0
-0.2
0
5
10
15
20
25
30
35
25
30
35
25
30
35
Generation Time
b)
Sensitivity of SER
to change in sub-adult survival
0.10
0.08
0.06
0.04
0.02
0.00
-0.02
-0.04
0
5
10
15
20
Generation Time
Sensitivity of SER
to change in adult survival
c)
0.00
-0.05
-0.10
-0.15
-0.20
-0.25
-0.30
0
5
10
15
20
Generation Time